You managed to capture Ditto chromatic in Pokémon GO and now you are moving on to your only Mew? Congratulations! Now a tough special quest awaits you. In all likelihood it will accompany you for a long time, so don't rush. It will remain available for you for a very long time.
Research "All for one, n. 151"Falls into the new category of Masterful Research. As the name suggests, they are aimed at demonstrating your skills as a Trainer, and as such they require an enormous amount of effort.
The first, at the end of which a chromatic Mew awaits you, is only available for those who have purchased a ticket for the Pokémon GO Tour of Kanto and have completed the search for Ditto.
Tips to complete
The research itself presents no particular difficulties. The most of the skills that are required is an enormous patience, especially for those missions that take time. The most difficult step (if you haven't already completed it) will surely be gathering the 151 Pokémon of Kanto, as both the legendaries and the region-locked have just left the raids. It is not known when they will be available again, but eyes open.
Catching one Pokémon a day for 30 days is in itself an easy mission, since even in country life situations you have one Pokémon guaranteed per day.
The same must be said for the friendship packs of the previous phase, as your partner at the second level of friendship will bring you some if you are left dry.
While you have a month of patience, take the opportunity to arm yourself with more patience and attention. They also serve you 151 different species of Pokémon to continue, so every catch opportunity is good (and you may want to mark who you have already caught if you want to help yourself).
In comparison it will be much easier gather 30 Pokémon of each type - especially since they don't have to be 30 different Pokémon. In this sense, the raid hours and the Pokémon in the foreground will be a godsend.